Gravity Penrose Process3 From Jcost

The Penrose process

The Penrose process is a way to extract energy from a rotating black hole, proposed by Roger Penrose in 1969. A particle entering the ergosphere, the region just outside the event horizon where spacetime is dragged along, can split into two. One part falls in, the other escapes with more energy than the original. The maximum efficiency of this extraction, for a maximally spinning black hole, is about 20.7 percent: you can get out at most that fraction of the infalling mass-energy.

The framework's library states a general result about its cost function, written J, which measures the price of a recognition event. For any positive numbers m and e, the cost J(m/e) is never negative, and it is zero exactly when m equals e. The library also proves that the golden ratio phi, about 1.618, is greater than 1.5, so the quantity phi minus 1.5 is positive. These three facts are machine-checked theorems, not assumptions.

In Recognition Science, the framework models the Penrose efficiency as eta = J(phi) times the ratio of the horizon angular velocity to the innermost stable circular orbit angular velocity. At maximal spin, that ratio is about 1.618, and J(phi) works out to about 0.118, giving an efficiency of about 19.1 percent. The measured maximum is 20.7 percent, so the framework's value is consistent with the known limit.

What the module itself proves is deliberately modest. Its definition of the cost for this process is just J(m/e), with no physical meaning attached to m and e. The three theorems hold for any positive inputs; they are general facts about J, not specific to black holes. The module records the intended application in a research note, but the note is not a theorem.

The useful consequence is that the framework's single cost function, derived from first principles, lands within the accepted efficiency window for a major astrophysical process. It does not derive the 20.7 percent figure; it reproduces it approximately. The gap between 19.1 and 20.7 percent remains a target for future work.

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module proves the 20.7 percent empirical maximum efficiency. This module derives the Penrose process efficiency from physical first principles. The cost function J is specific to black hole physics.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/PenroseProcess3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: